Simulating a screw turning a gear
Animating gears is usually done using the Copy Rotation  constraint, in your case you want to copy the rotation from another axis therefore you might want to use the Transformation constraint to copy and scale the rotation from one axis to another:





Video Tutorial:
Gear rigging tutorial constraints

Note: This is only to do an animation, I'm not sure whether Blender is suitable for simulation of mechanical parts from an engineering point of view.